P. Jeyanthi and N. Angel Benseera

Transcription

1 Opuscula Math. 34, no. 1 (014), Opuscula Mathematica A TOTALLY MAGIC CORDIAL LABELING OF ONE-POINT UNION OF n COPIES OF A GRAPH P. Jeyanthi and N. Angel Benseera Communicated by Dalibor Fronček Abstract. A graph G is said to have a totally magic cordial (TMC) labeling with constant C if there exists a mapping f : V (G) E(G) 0, 1} such that f(a)+f(b)+f(ab) C(mod ) for all ab E(G) and n f (0) n f (1) 1, where n f (i) (i = 0, 1) is the sum of the number of vertices and edges with label i. In this paper, we establish the totally magic cordial labeling of one-point union of n-copies of cycles, complete graphs and wheels. Keywords: totally magic cordial labeling, one-point union of graphs. Mathematics Subject Classification: 05C INTRODUCTION All graphs considered here are finite, simple and undirected. The set of vertices and edges of a graph G is denoted by V (G) and E(G) respectively. Let p = V (G) and q = E(G). A general reference for graph theoretic ideas can be seen in [3]. The concept of cordial labeling was introduced by Cahit [1]. A binary vertex labeling f : V (G) 0, 1} induces an edge labeling f : E(G) 0, 1} defined by f (uv) = f(u) f(v). Such labeling is called cordial if the conditions v f (0) v f (1) 1 and e f (0) e f (1) 1 are satisfied, where v f (i) and e f (i) (i = 0, 1) are the number of vertices and edges with label i respectively. A graph is called cordial if it admits a cordial labeling. The cordiality of a one-point union of n copies of graphs is given in [6]. Kotzig and Rosa introduced the concept of edge-magic total labeling in [5]. A bijection f : V (G) E(G) 1,, 3,..., p + q} is called an edge-magic total labeling of G if f(x) + f(xy) + f(y) is constant (called the magic constant of f) for every edge xy of G. The graph that admits this labeling is called an edge-magic total graph. The notion of totally magic cordial (TMC) labeling was due to Cahit [] as a modification of edge magic total labeling and cordial labeling. A graph G is said to have TMC labeling with constant C if there exists a mapping f : V (G) E(G) 0, 1} c AGH University of Science and Technology Press, Krakow

2 116 P. Jeyanthi and N. Angel Benseera such that f(a) + f(b) + f(ab) C(mod ) for all ab E(G) and n f (0) n f (1) 1, where n f (i) (i = 0, 1) is the sum of the number of vertices and edges with label i. A rooted graph is a graph in which one vertex is named in a special way so as to distinguish it from other nodes. The special node is called the root of the graph. Let G be a rooted graph. The graph obtained by identifying the roots of n copies of G is called the one-point union of n copies of G and is denoted by G (n). In this paper, we establish the TMC labeling of a one-point union of n-copies of cycles, complete graphs and wheels.. MAIN RESULTS In this section, we present sufficient conditions for a one-point union of n copies of a rooted graph to be TMC and also obtain conditions under which a one-point union of n copies of graphs such as a cycle, complete graph and wheel are TMC graphs. We relate the TMC labeling of a one-point union of n copies of a rooted graph to the solution of a system which involves an equation and an inequality. Theorem.1. Let G be a graph rooted at a vertex u and for i = 1,,..., k, f i : V (G) E(G) 0, 1} be such that f i (a) + f i (b) + f i (ab) C(mod ) for all ab E(G) and f i (u) = 0. Let n fi (0) = α i, n fi (1) = β i for i = 1,,..., k. Then the one-point union G (n) of n copies of G is TMC if the system (.1) has a nonnegative integral solution for the x i s: k k k (α i 1)x i β i x i and x i = n. (.1) i=1 i=1 Proof. Suppose x i = δ i, i = 1,,..., k, is a nonnegative integral solution of system (.1). Then we label the δ i copies of G in G (n) with f i (i = 1,,..., k). As each of these copies has the property f i (a) + f i (b) + f i (ab) C(mod ) and f i (u) = 0 for all i = 1,,..., k, G (n) is TMC. Corollary.. Let G be a graph rooted at a vertex u and f be a labeling such that f(a)+f(b)+f(ab) C(mod ) for all ab E(G) and f(u) = 0. If n f (0) = n f (1)+1, then G (n) is TMC for all n 1. Example.3. One point union of a path is TMC. Corollary.4. Let G be a graph rooted at u. Let f i, i = 1,, 3 be labelings of G such that f i (a) + f i (b) + f i (ab) C(mod ) for all ab E(G), f i (u) = 0 and γ i = α i β i. 1. If γ 1 = and γ =, then G (n) is TMC for all n 1(mod 4).. If either a) γ 1 = 1 and γ = 3, or b) γ 1 = 4, γ = and γ 3 = 4, or c) γ 1 = 3, γ = 3 and γ 3 = 5, then G (n) is TMC for all n If γ 1 = 0 and γ = 4, then G (n) is TMC for all n 3(mod 4). i=1

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